Archive for recycling

unbiased product of expectations

Posted in Books, Statistics, University life with tags , , , , , , , , on August 5, 2019 by xi'an

m_biomet_106_2coverWhile I was not involved in any way, or even aware of this research, Anthony Lee, Simone Tiberi, and Giacomo Zanella have an incoming paper in Biometrika, and which was partly written while all three authors were at the University of Warwick. The purpose is to design an efficient manner to approximate the product of n unidimensional expectations (or integrals) all computed against the same reference density. Which is not a real constraint. A neat remark that motivates the method in the paper is that an improved estimator can be connected with the permanent of the n x N matrix A made of the values of the n functions computed at N different simulations from the reference density. And involves N!/ (N-n)! terms rather than N to the power n. Since it is NP-hard to compute, a manageable alternative uses random draws from constrained permutations that are reasonably easy to simulate. Especially since, given that the estimator recycles most of the particles, it requires a much smaller version of N. Essentially N=O(n) with this scenario, instead of O(n²) with the basic Monte Carlo solution, towards a similar variance.

This framework offers many applications in latent variable models, including pseudo-marginal MCMC, of course, but also for ABC since the ABC posterior based on getting each simulated observation close enough from the corresponding actual observation fits this pattern (albeit the dependence on the chosen ordering of the data is an issue that can make the example somewhat artificial).

plastic oceans

Posted in Kids, pictures with tags , , , , , , , , on May 28, 2018 by xi'an

conference carbon footprint

Posted in Kids, pictures, Running, Travel, University life, Wines with tags , , , , , , , , , on August 1, 2017 by xi'an

As a local organiser of the recent BNP 11 conference in Paris, and hence involved in setting and cleaning coffee breaks and [now famous] wine&cheese poster sessions, I was rather shocked by the amount of waste generated by those events, albeit aware of the importance of the social exchanges they induced… And thus got to wonder how the impact of those conference events could be reduced. One solution is the drastic one, namely to provide exactly nothing at all during the breaks between talks and expect anyone hungry or thirsty enough to bring one own’s food or drink. Another one, as suggested by my daughter at the dinner table, is to provide Ecocups, namely reusable plastic glasses that can given to all participants at the beginning of the conference. Or sold (or rented) to those who have not brought their own mug or bottle. (Of course, this may be a poor idea in that manufacturing and shipping a hard-plastic glass that most likely will be discarded after a few days may be more damaging than producing the equivalent number of “disposable” thin plastic glasses. And in the end all this agitation is peanuts compared with the impact of flying participants to the conference. For which I have no handy solution… As biking to the conference location is a privilege very few can enjoy.) Still, and even though this puts another stone in the already rocky organisers’ garden, I wish we could adopt more positive policies at the meetings we organise and sponsor.

impressions from EcoSta2017 [guest post]

Posted in pictures, Statistics, Travel, University life with tags , , , , , , , , , on July 6, 2017 by xi'an

[This is a guest post on the recent EcoSta2017 (Econometrics and Statistics) conference in Hong Kong, contributed by Chris Drovandi from QUT, Brisbane.]

There were (at least) two sessions on Bayesian Computation at the recent EcoSta (Econometrics and Statistics) 2017 conference in Hong Kong. Below is my review of them. My overall impression of the conference is that there were lots of interesting talks, albeit a lot in financial time series, not my area. Even so I managed to pick up a few ideas/concepts that could be useful in my research. One criticism I had was that there were too many sessions in parallel, which made choosing quite difficult and some sessions very poorly attended. Another criticism of many participants I spoke to was that the location of the conference was relatively far from the city area.

In the first session (chaired by Robert Kohn), Minh-Ngoc Tran spoke about this paper on Bayesian estimation of high-dimensional Copula models with mixed discrete/continuous margins. Copula models with all continuous margins are relatively easy to deal with, but when the margins are discrete or mixed there are issues with computing the likelihood. The main idea of the paper is to re-write the intractable likelihood as an integral over a hypercube of ≤J dimensions (where J is the number of variables), which can then be estimated unbiasedly (with variance reduction by using randomised quasi-MC numbers). The paper develops advanced (correlated) pseudo-marginal and variational Bayes methods for inference.

In the following talk, Chris Carter spoke about different types of pseudo-marginal methods, particle marginal Metropolis-Hastings and particle Gibbs for state space models. Chris suggests that a combination of these methods into a single algorithm can further improve mixing. Continue reading

locally weighted MCMC

Posted in Books, Statistics, University life with tags , , , , , , , , on July 16, 2015 by xi'an

Street light near the St Kilda Road bridge, Melbourne, July 21, 2012Last week, on arXiv, Espen Bernton, Shihao Yang, Yang Chen, Neil Shephard, and Jun Liu (all from Harvard) proposed a weighting scheme to associated MCMC simulations, in connection with the parallel MCMC of Ben Calderhead discussed earlier on the ‘Og. The weight attached to each proposal is either the acceptance probability itself (with the rejection probability being attached to the current value of the MCMC chain) or a renormalised version of the joint target x proposal, either forward or backward. Both solutions are unbiased in that they have the same expectation as the original MCMC average, being some sort of conditional expectation. The proof of domination in the paper builds upon Calderhead’s formalism.

This work reminded me of several reweighting proposals we made over the years, from the global Rao-Blackwellisation strategy with George Casella, to the vanilla Rao-Blackwellisation solution we wrote with Randal Douc a few years ago, both of whom also are demonstrably improving upon the standard MCMC average. By similarly recycling proposed but rejected values. Or by diminishing the variability due to the uniform draw. The slightly parallel nature of the approach also connects with our parallel MCM version with Pierre Jacob (now Harvard as well!) and Murray Smith (who now leaves in Melbourne, hence the otherwise unrelated picture).

recycling accept-reject rejections

Posted in Statistics, University life with tags , , , , , , , , , on July 1, 2014 by xi'an

Vinayak Rao, Lizhen Lin and David Dunson just arXived a paper which proposes anew technique to handle intractable normalising constants. And which exact title is Data augmentation for models based on rejection sampling. (Paper that I read in the morning plane to B’ham, since this is one of my weeks in Warwick.) The central idea therein is that, if the sample density (aka likelihood) satisfies

p(x|\theta) \propto f(x|\theta) \le q(x|\theta) M\,,

where all terms but p are known in closed form, then completion by the rejected values of an hypothetical accept-reject algorithm−hypothetical in the sense that the data does not have to be produced by an accept-reject scheme but simply the above domination condition to hold−allows for a data augmentation scheme. Without requiring the missing normalising constant. Since the completed likelihood is

\prod_{i=1}^n \dfrac{f(x_i|\theta)}{M} \prod_{j=1}^{m_i} \left\{q(y_{ij}|\theta) -\dfrac{f(y_{ij}|\theta)}{M}\right\}

A closed-form, if not necessarily congenial, function.

Now this is quite a different use of the “rejected values” from the accept reject algorithm when compared with our 1996 Biometrika paper on the Rao-Blackwellisation of accept-reject schemes (which, still, could have been mentioned there… Or Section 4.2 of Monte Carlo Statistical Methods. Rather than re-deriving the joint density of the augmented sample, “accepted+rejected”.)

It is a neat idea in that it completely bypasses the approximation of the normalising constant. And avoids the somewhat delicate tuning of the auxiliary solution of Moller et al. (2006)  The difficulty with this algorithm is however in finding an upper bound M on the unnormalised density f that is

  1. in closed form;
  2. with a manageable and tight enough “constant” M;
  3. compatible with running a posterior simulation conditional on the added rejections.

The paper seems to assume further that the bound M is independent from the current parameter value θ, at least as suggested by the notation (and Theorem 2), but this is not in the least necessary for the validation of the formal algorithm. Such a constraint would pull M higher, hence reducing the efficiency of the method. Actually the matrix Langevin distribution considered in the first example involves a bound that depends on the parameter κ.

The paper includes a result (Theorem 2) on the uniform ergodicity that relies on heavy assumptions on the proposal distribution. And a rather surprising one, namely that the probability of rejection is bounded from below, i.e. calling for a less efficient proposal. Now it seems to me that a uniform ergodicity result holds as well when the probability of acceptance is bounded from below since, then, the event when no rejection occurs constitutes an atom from the augmented Markov chain viewpoint. There therefore occurs a renewal each time the rejected variable set ϒ is empty, and ergodicity ensues (Robert, 1995, Statistical Science).

Note also that, despite the opposition raised by the authors, the method per se does constitute a pseudo-marginal technique à la Andrieu-Roberts (2009) since the independent completion by the (pseudo) rejected variables produces an unbiased estimator of the likelihood. It would thus be of interest to see how the recent evaluation tools of Andrieu and Vihola can assess the loss in efficiency induced by this estimation of the likelihood.

Maybe some further experimental evidence tomorrow…